University of Florida/Egm4313/s12.team6.hill/R1

Problem 1: Spring-dashpot system in parallel

Derive the equation of motion of a spring-dashpot system in parallel, with a mass and applied force $f(t)$

Given

Spring-dashpot system in parallel

Solution

The kinematics of the system can be described as,

$\displaystyle x=x_{k}=x_{c}$

$\longrightarrow (1)$

The kinetics of the system can be described as,

$\displaystyle m{\ddot {x}}+f_{I}=f(t)$

$\longrightarrow (2)$

and,

$\displaystyle f_{I}=f_{k}+f_{c}$

$\longrightarrow (3)$

Given that,

$\displaystyle f_{k}=kx_{k}$

$\longrightarrow (3a)$

$\displaystyle f_{c}=c{\dot {x_{c}}}$

$\longrightarrow (3b)$

From (1), it can be found that,

$\displaystyle {\dot {x}}={\dot {x_{k}}}={\dot {x_{c}}}$

and,

$\displaystyle {\ddot {x}}={\ddot {x_{k}}}={\ddot {x_{c}}}$

From (3), it can be found that,

$\displaystyle f_{I}=c{\dot {x_{c}}}+kx_{k}$

Finally, it can be found that

$\displaystyle m{\ddot {x}}+c{\dot {x}}+kx=f(t)$